#import "SimpleTem.typ" : conf
#show: doc => conf(
  [Well-Balanced and Positivity-Preserving for One Dimensional Euler equation with Gravity],[WB and PP], authors:
    (
      (name: "Yu Wang",
      affiliation: "ECNU",
      )
    ),
  doc,
)
= Introduction
We consider the one dimensional Euler equations with gravity source term:
$ bold(w)_t + bold(f)(bold(w))_x = bold(s)(bold(w),x) $
with
$ bold(w) = vec(rho, m, E), "   "bold(f)(bold(w)) = vec(rho u, rho u^2 + p, u(E + p)), "  " bold(s) = vec(0, -rho phi.alt_x, -m phi.alt_x) $
In the above system, $rho$ is the fluid density, $u$ is the velocity, $m = rho u$ is the momentum, $E$ is the non-gravitational energy which is the summation of the kinetic energy and internal energy of the fluid, and $p$ is the pressure. The source term $bold(s)$ represents the effect of gravitational field with $phi.alt =phi.alt(x)$ being the time independent gravitational potential. To close the system, we consider the ideal gas law
$ p = (gamma -1)(E - 1/2 rho u^2) $ 
where $gamma > 1$ is the ratio of specific heats. System with the ideal gas law admits steady-state soltions ${rho^e (x), p^e (x), u^e (x)}$ which are obtained by exact balance of the flux and the source term:
$ u^e (x) = 0 , "  "(p^e (x))_x = - rho^e (x) phi.alt_x $
= One dimensional problems
== Numerical schemes